Dimensional reduction in networks of
non-Markovian spiking neurons: Equivalence of
synaptic filtering and heterogeneous
propagation delays
Maurizio MattiaID1*, Matteo Biggio2, Andrea Galluzzi1, Marco StoraceID2
1 Istituto Superiore di Sanità, Roma, Italy, 2 DITEN, University of Genoa, Genova, Italy *maurizio.mattia@iss.it
Abstract
Message passing between components of a distributed physical system is non-instanta-neous and contributes to determine the time scales of the emerging collective dynamics. In biological neuron networks this is due in part to local synaptic filtering of exchanged spikes, and in part to the distribution of the axonal transmission delays. How differently these two kinds of communication protocols affect the network dynamics is still an open issue due to the difficulties in dealing with the non-Markovian nature of synaptic transmission. Here, we develop a mean-field dimensional reduction yielding to an effective Markovian dynamics of the population density of the neuronal membrane potential, valid under the hypothesis of small fluctuations of the synaptic current. Within this limit, the resulting theory allows us to prove the formal equivalence between the two transmission mechanisms, holding for any synaptic time scale, integrate-and-fire neuron model, spike emission regimes and for differ-ent network states even when the neuron number is finite. The equivalence holds even for larger fluctuations of the synaptic input, if white noise currents are incorporated to model other possible biological features such as ionic channel stochasticity.
Author summary
Understanding the collective behavior of the intricate web of neurons composing a brain is one of the most challenging and complex tasks of modern neuroscience. Part of this complexity resides in the distributed nature of the interactions between the network com-ponents: for instance, the neurons transmit their messages (through spikes) with delays, which are due to different axonal lengths (i.e., communication distances) and/or non-instantaneous synaptic transmission. In developing effective network models, both of these aspects have to be taken into account. In addition, a satisfactory description level must be chosen as a compromise between simplicity and faithfulness in reproducing the system behavior. Here we propose a method to derive an effective theoretical description —validated through network simulations at microscopic level—of the neuron population dynamics in many different working conditions and parameter settings, valid for any a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 OPEN ACCESS
Citation: Mattia M, Biggio M, Galluzzi A, Storace M
(2019) Dimensional reduction in networks of non-Markovian spiking neurons: Equivalence of synaptic filtering and heterogeneous propagation delays. PLoS Comput Biol 15(10): e1007404.
https://doi.org/10.1371/journal.pcbi.1007404 Editor: Bard Ermentrout, University of Pittsburgh,
UNITED STATES
Received: May 2, 2018 Accepted: September 16, 2019 Published: October 8, 2019
Copyright:© 2019 Mattia et al. This is an open access article distributed under the terms of the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability Statement: All relevant data are
within the paper and its Supporting Information files.
Funding: Work partially supported by the EU
Horizon 2020 Research and Innovation
Programme under HBP SGA2 (grant no. 785907 to MM) and by the University of Genoa (MS). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
synaptic time scale. In doing this we assume relatively small instantaneous fluctuations of the input synaptic current. As a by-product of this theoretical derivation, we prove analyt-ically that a network with non-instantaneous synaptic transmission with fixed spike deliv-ery delay is equivalent to a network characterized by a suited distribution of spike delays and instantaneous synaptic transmission, the latter being easier to treat.
Introduction
Large distributed systems like brain neuronal networks often have to satisfy both timing and space constraints, irrespective of their size. Indeed, they have to be compact in order to be por-table, and at the same time the neurons have to communicate always to the same pace, in part determined by the environment [1]. Communication through spikes makes this possible in a distributed highly-parallel architecture with low power density. Spike communication in these networks relies on both a synaptic low-pass filtering and a suited distribution of axonal propa-gation delays: a heterogeneous transmission mechanism, which in turn affects the collective dynamics of the network. As a consequence, even in the simplest modeling condition of point-like spiking neurons, the microscopic dynamics of their state variable (i.e., the membrane
potentialV(t)) is non-Markovian, and hence multi-dimensional through Markovian
embed-ding [2,3]. This microscopic dynamics spans a wide variety of time scales, affecting the stabil-ity of various network dynamics [4], the selectivity in transmitting information [5] and the reactivity to suddenly appearing exogenous stimuli [6].
Despite the long history of attempts in statistical physics to work out an effective dimen-sional reduction leading to a Markovian description of the dynamics of a non-Markovian sys-tem [2,7], a theoretical framework valid for any correlation time of the synaptic input and any neuronal activity regime is still missing. Indeed, theoretical approaches including non-instan-taneous transmission rely on approximations valid only for relatively small time scales [6,8–
10], or for quasi-adiabatic dynamical regimes [11,12], or for neurons working with low-noise supra-threshold inputs [13,14].
Here, we address this issue by developing a theoretical framework in which from small to large synaptic time scales the same dynamical model holds for the instantaneous firing rate of spiking neuron networks. Starting from the population density dynamics of single-neuron state variables under mean-field approximation, we extend to the colored-noise case the spec-tral expansion of the associated Fokker-Planck (FP) equation previously derived under white-noise assumption [15]. Resorting to a kind of central moment closure method [16], a zero-th order approximation of the population density dynamics is worked out, recovering the same theoretical framework as for the white-noise case. In this framework, the projections of the population density onto the basis composed of the non-stationary eigenmodes of the FP opera-tor (i.e., the eigenfunctions with non-zero eigenvalues) are used to fully represent the state of the network. This basis moves driven by the time-varying moments of the synaptic current which depends on a low-pass filtered version of the network spiking activity. In this way, the basis effectively adapts to the population density evolution and the network dynamics is described by a compact system, valid for a wide class of models and dynamical regimes. We validate the effectiveness of this theoretical description showing a remarkable agreement with microscopic simulations of networks with various integrate-and-fire (IF) neuron models (leaky, exponential and VLSI), spike emission regimes (sub- and supra-threshold) and dynamical states (asynchronous states, corresponding to stable equilibrium points of the Competing interests: The authors have declared
instantaneous firing rate and oscillating synchronous states, corresponding to limit cycles), even when finite-size fluctuations are taken into account.
Finally, we prove a formal equivalence between the two transmission mechanisms: a suited distribution of spike transmission delays between neurons allows to fully reproduce the firing rate dynamics of the same network having instead non-instantaneous synaptic transmission. It is then not by chance that both strategies coexist in a huge cellular network like the one com-posing our brain. Indeed, thanks to such interchangeability, the balance between these mecha-nisms might be optimally tuned to satisfy both compactness and communication constraints.
Results
Theoretical results
Full population density model as a set of interacting Markovian systems. We consider
a network composed ofN integrate-and-fire (IF) neurons, each receiving an input current I(t) resulting from the low-pass filtered linear combination of the spikes emitted at timestj, with
Poissonian distribution, by a subset of presynaptic cells in the network [9,17]. For simplicity, here we assume a first-order dynamics for synaptic currentI, with decay time τsand constant
efficacyJ : ts_I ¼ I þ tmJ P
jdðt tjÞ=R. The membrane potential V(t) of an IF neuron
evolves according to the general equation tmV ¼_ f ðVÞ þ R ðI þ IextÞ, with membrane decay
constantτm, neuron resistanceR = τm/Cm, membrane capacitanceCmand a
voltage-depen-dent leakage driftf(V), which depends on the model neuron type. For instance, f(V) is a con-stant drift for the perfect IF (PIF) neuron andf(V) = V for the leaky IF (LIF) [18,19].
Neurons may receive an additional currentIextmodeling external sources, like incoming
synaptic input from other networks. OnceV crosses a threshold value vthr, a spike is emitted
and potentialV is reset to vres<vthr(boundary conditions).Fig 1Ashows a Poisson spike
train (top panel) and the corresponding input currentI(t) (middle panel) and membrane potentialV(t) (bottom panel) for a LIF neuron with τs= 4 ms,τm= 10 ms andIext= 0.
Due to the boundary conditions, even under stationary regimes single-neuron dynamics is not analytically tractable. This is particularly apparent looking at the stationary probability dis-tribution computed from the long time series partially shown inFig 1A. The probability distri-bution of states (V, I) worked out numerically is shown inFig 1B. The absorbing barrier invthr
and the reentering flux of realizations invresmodeling the emission of spikes make this
distri-bution asymmetric [6,8,20] and correlation betweenV and I non-monotonic, a feature even more apparent for slower synaptic filtering, as shown inFig 1C and 1D, obtained by increasing τsto 64 ms.
Under diffusion approximation, holding for large rate of incoming spikes each only mildly affectingV [18,19], membrane potential dynamics is described by the following system of Langevin equations [2]
tmV ¼_ f ðVÞ þ RðI þ IextÞ ; ts_I ¼ I þ
mIþ sIx
R ; ð1Þ
whereRIext=μext+σextξextis a Gaussian white noise. We remark that the filtering effect of
synaptic dynamics makes the noise colored, thus leading to finite correlation time constants. Moreover,V(t), individually, is not Markovian but V(t) and I(t), together, constitute a bi-vari-ate Markov process [2].
Fig 2provides an at-a-glance description of our method to obtain the final population dynamics and is used as a main reference throughout this section. The first intermediate Lan-gevin model (1) corresponds toFig 2A.
When the mean driving force alone (i.e., the deterministic part of the inputI + Iext) is not
enough to make the membrane potentialV cross the threshold vthr, the neurons are evolving
in anoise-dominated (or subthreshold) regime and irregular firing occurs, due to the
sto-chastic part of the input. In the opposite case, the emission of an action potential can occur also in the absence of noisy afferent currents, and the neurons are in adrift-dominated (or suprathreshold)regime of activity, characterized by the regular emission of spike trains [21–23].
The stochastic differentialEq (1)describe a single neuron where the time course ofV(t) andI(t) is probabilistic. Therefore, the dynamics of V and I is fully described by the probability densityρ(V, I, t).Fig 1B and 1Dare examples of distributions of (V, I) samples obtained from the numerical integration of the single-neuron dynamics under stationary conditions and before assuming the diffusion approximation. The probability densityρ(V, I, t) describes (under diffusion approximation) how this distribution evolves in time.
If we consider a network of statistically identical neurons, each with a different realization of the stochastic input,ρ(V, I, t) can be considered a population density. The population den-sity approach [2,24] provides a method to analyze the dynamics of ensembles of connected neurons, relating the firing properties of a network to the features of single neurons and of their synaptic connections. This is done by assuming the “mean-field” hypothesis, which allows to use the probability density for independent neurons in a “self-consistent” way. Fig 1. Spike input filtering by synaptic transmission. (A, C) Presynaptic activity modeled by a Poissonian spike train
(top) is low pass-filtered by the synaptic transmission to produce the input currentI(t) (middle) to the membrane potentialV(t) (bottom) of a leaky integrate-and-fire (LIF) neuron. Examples are shown for fast (A, τs= 4 ms) and slow
(C,τs= 64 ms) synaptic filtering. (B, D) Stationary distribution in the plane (V, I) sampled from simulations of the
same LIF neurons as in A and C with fast (B) and slow (D) synaptic transmission. Red arrows: mean fluxes of realizations (probability currents) at different (V, I). vthr: absorbing barrier representing the spike emission threshold. vres: reset membrane potential following the emission of a spike. HereIext= 0.
According to this theoretical ansatz, the infinitesimal moments ofRI mIðtÞ ¼ tmJ C nðtÞ s2 IðtÞ ¼ tmJ 2C nðtÞ ð2Þ
characterize the white noiseξ driving the activity-dependent synaptic current [8,25] as a func-tion of the network firing rateν(t) (i.e., the average number of spikes each neuron emits per time unit) and the average number of presynaptic contactsC (� N) per neuron. The probabil-ity densprobabil-ityρ(x, y, t) to find neurons at time t with membrane potential V(t) = x and synaptic currentI(t) = y/R follows the FP equation [2] (seeFig 2B):
@tr ¼ @xFxðx; yÞ @yFyðy; nÞ � 1 tm @x ðf ðxÞ y mextÞ þ 1 2s 2 ext@x � � rþ 1 ts@y ðy mIÞ þ 1 2s 2 I@y h i r � ðLxþLyÞr : ð3Þ Fig 2. Sketch of the main steps of the theoretical derivation. (A) Diffusion approximation (right) of the single-neuron dynamics.
(B) Probability densityρ(x, y) at fixed t, with the associated marginal distributions for both membrane potential (ρx, gray) and
current (ρy, orange). Orange dashed line: instantaneous meanμyof the current (see main text for details). (C) “Slices” of the
two-dimensionalρ(x, y) bounded by red dotted lines; each “slice” (which actually has infinitesimal thickness) corresponds to a one-dimensional subpopulation of neurons with similary and is described by the expansion coefficients {an(y)} of a suited
one-dimensional FP operator. (D) Partial firing ratesνy(purple) are integrated overy to obtain the ν(t) of the whole network. (E) The
instantaneous firing rateν(t) is now the only state variable needed to be fed back in order to operate the moment closure and take into account only the “slice” centered aroundμy(the 0-th order approximation).
This is a continuity equation in which the density changes are determined by the divergence of the probability current ðFx;FyÞ (i.e., the flux of realizations) together with few specific bound-ary conditions (seeFig 1): i) spike emission as an absorbing barrier atx = vthr(only fory > 0
ifσext= 0), ii) reentering flux of absorbed realizations at the reset potentialvresand iii) lower
bound for the membrane potential implemented as a reflecting barrier atvmin�vres[8,15,
26,27].
In general,Eq (3)is hard to solve and only approximated solutions (mainly under station-ary conditions) have been worked out [2,6,7,28]. Here we derive a solution for this two-dimensional problem without making any assumption like stationarity in time and restricted ranges for the synaptic filtering timescale. We extend the spectral expansion approach exploited to study the one-dimensional case [15,26], by describingρ(x, y, t) as the superposi-tion of aninfinite set of one-dimensional interacting sub-populations; each sub-population col-lects the neurons that at timet receive a synaptic current y/R. Therefore we have iso-current sub-populations labeled by y and changing in time as the realizations move along the y-dimen-sion due to the action ofFy. In other words, we “slice” the two-dimensional diagrams of the probability current ðFx;FyÞ corresponding toρ(x, y, t) along the y direction, thus obtaining probability densities rðx; y; tÞjy¼�y, as sketched inFig 2C.
The dynamics of each iso-current sub-population is studied, as in [15], by projectingρ at fixedy on the eigenfunctions {|ϕmi} of a suited one-dimensional FP operator, which here we
set to be Lxy ¼ 1 tm@x½f ðxÞ y mext� þ 1 2tm s 2 extþ s 2 ðyÞ � � @2x; ð4Þ
with an additional diffusion termσ2(y) with respect to LxofEq (3), which is the variance of an
arbitrary Gaussian white noise depending on the state variable y.
As shown later, this additional term will play a crucial role in our dimensional reduction. Indeed,σ2(y) will be chosen self-consistently to take into account the statistical variability of the input currents received by all the iso-current subpopulations. This unconventional choice allows us to use an expansion in eigenfunctions of this FP operator, with coefficients anðy; tÞ ¼ hcnjri ¼
Rv
thr
vmincnðx; yÞ rðx; y; tÞ dx, which now explicitly depend on the term y
and where hψn| is then-th eigenfunction of the adjoint operator of Lxy. The stationary
mode with eigenvalueλ0= 0 has hψ0| = 1, thusa0(y, t) is the marginal probability density
ryðy; tÞ ¼Rvthr
vminrðx; y; tÞ dx (Fig 2, orange distribution).
The dynamics ofancan be worked out by derivinganwith respect to time [15,26], and
rewritingLx¼Lxy 1 2tms 2ðyÞ@2 x: _an ¼ hcnjðLxþLyÞri ¼ hcnjLxri þ hcnjLyri ¼ ¼ hcnjLxyri s 2ðyÞ 2tm hcnj@ 2 xri þ hcnjLyri ¼ ¼ lnan s2ðyÞ 2tm X q aqhcnj@ 2 x�qi þ X q hcnjLyaq�qi : ð5Þ
where |ϕqi and hψn| are the eigenfunctions of the operatorLxyand of its adjoint, respectively.
depend onx: _ry¼Lyry rys 2ðyÞ 2tm Z v thr vmin @2x�0ðx; yÞ dx :
The integral on the r.h.s. can be solved by parts and turns out to be 0 thanks to both the conservation of the flux of realizations exiting fromvthrand re-entering invres
(@x�0jx¼vthr ¼ @x�0jx¼vresþ @x�0jx¼vres ) and the reflecting barrier condition invmin, provided
that the reasonable assumptionϕ0(vmin,y) = 0 holds (this is surely true for vmin!−1, as ϕ0is
a probability density). Thus, theρydynamics reduces to
_ry¼Lyry¼ @yðy mIÞry ts þ s2 I 2ts@ 2 yry: ð6Þ
Note that this expression can be derived directly from the FPEq (3).
Relying on the same spectral expansion as in [15], each ‘partial’ rate nyðy; tÞ ¼ Fxðvthr;yÞ (Fig 2, purple distribution) can be decomposed as follows:
nyðy; tÞ ¼ ryðy; tÞ Fðy; nÞ þ
X
n6¼0
anðy; tÞ fnðy; nÞ ; ð7Þ
where the stationary mode contribution is separated from the others. The network firing rate ν(t) is obtained by integrating the partial rates νy(y, t) over y, as sketched inFig 2D:
nðtÞ ¼ Z 1 1 Fxðvthr;yÞdy � Z 1 1 nyðy; tÞdy ; ð8Þ
In this framework, Fðy; nÞ � Fxðvthr;yÞjr¼�0is the firing rate of the stationary
one-dimen-sional densityϕ0at fixedy, while the other nonstationary modes contribute with the fluxes
fnðy; nÞ � Fxðvthr;yÞjr¼�n. Both F andfndepend on the total firing rateν(t) through the input
current momentsμIandσI. The main advantage of having transformed the original
two-dimensional description into a set of infinite one-two-dimensional FP equations is that the dynam-ics described by these equations for each iso-frequency population is known to be tractable for a wide range of network settings. This is crucial to derive an approximated dynamics ofν(t) valid for any synaptic time scaleτs, neuron model and dynamical regime, as shown in the
following.
Dimensional reduction of the firing rate dynamics. Eqs(7)and(8)together are a reformulation of the original problem (3), not a solution. However, in this framework a pertur-bative approach can be envisaged. FromEq (6), synaptic currenty results to have time-depen-dent meanμy(t) = hyi and variance s2yðtÞ ¼ h½y myðtÞ�
2 i ts _my¼ myþ mI; ts_s2 y¼ 2s 2 yþ s 2 I; ð9Þ
like an Ornstein-Uhlenbeck process with nonstationary input momentsμI(t) and σI(t) [2]. We
remark that s2
yshould not be confused with the arbitrary and additional diffusion termσ
2
(y) introduced in the definition ofLxy: s2
yis the variance of the synaptic currenty, whereas σ
2
(y) is the state-dependent variance of an unknown diffusion process included inLxy, which will be chosen later to obtain a self-consistent equation for the firing rateν(t).
Synaptic current displacement in time is jy myj ¼OðsIÞ and, for smallσI,—i.e., forσI�
vthr– we can assume a negligible role for ay distant enough from μy. This assumption of a
nar-rowρycentered aroundy = μyallows to simplify the integrals across they domain weighted by
by expanding in Taylor’s series they-dependent functions in the integrands, as detailed in the
S1 Appendixand sketched inFig 2E.
The next step is to find a self-consistent approximation (neglecting terms of orderOðsIÞ) ofν(t) in terms of 0-th order functions F0(ν) � F(μy,ν) and f0n(ν) � fn(μy,ν) and of
corre-sponding coefficients:
n � F0ðnÞ þ
X
n6¼0
a0nf0nðnÞ: ð10Þ
The dynamics of then-th expansion coefficient is ruled by the following equation, neglect-ing again terms of orderOðsIÞ:
_a0n� l0na0nþ _my
X
q
h@ycnj�qijy¼m
ya0q; ð11Þ
whereλ0n�λn(μy) (see details in theS1 Appendix).
This is a dimensional reduction ofEq (5)as the coefficientsa0nðtÞ �
R1
1anðy; tÞ dy do not
depend ony, and it holds provided that y has a narrow distribution across neurons at each timet (i.e., small σI(t)), leaving τsunconstrained.
To obtain a self-consistent equation for the firing rateν(t), we still need to choose a suited σ(y) in Lxy. To this purpose, the case of instantaneous synaptic transmissionτs= 0 can be used
as reference, asEq (3)reduces to a one-dimensional FP equation with the only operator Lx0¼ 1=tm@x½mIþ mext f ðxÞ� þ ðs2 extþ s 2 IÞ=ð2tmÞ@ 2 x[15,21,27]. Hence,Lxyjy¼my
must tend toLx0for vanishingτs, which is the case ifσ2(y) = Jy. Indeed, in this limit
s2ðm
yÞ ¼ tmJ
2C n ¼ s2
I, beingμy=μI=τmJ C ν fromEq (9). This implies that eigenvaluesλ0n
and eigenfunctions inEq (11)are the same as in the one-dimensional case withδ-correlated synaptic input, but with collective firing rateν(t) = μy(t)/(τmJC).
To summarize, s2
Iis the variance of the white noise driving the synaptic current and
depends onν(t) (seeEq (2)); s2
extis the variance of the Gaussian white noise representing the
external input (seeEq (1)); s2
yis the variance of the synaptic currenty (seeEq (9));σ
2
(y) = Jy is the variance of the diffusion process appearing inLxy(seeEq (4)), which is different for each iso-current “slice” being dependent ony. However, in the limit τs! 0 (where the decay time
τsis the relaxation time ofμy(t), seeEq (9)), the synaptic transmission becomes instantaneous
and s2
y ! s
2
I=2, differently from the varianceσ
2
(y) which in the relevant iso-current “slice” at y = μyis s2I.
In conclusion, this dimensional reduction extends the firing rate equation previously worked out forτs= 0 [15], and, by combining Eqs(9)–(11)and neglectingOðsIÞ terms, it
results to be _ ~a0¼ ðΛ0þW0 _myÞ~a0þ ~w0 _my _my¼ ðmIðnÞ myÞ=ts n ¼ F0þ ~f0� ~a0 : 8 > > > > < > > > > : ð12Þ
Here, the infinite vectors ~a0 ¼ fa0ng and ~f0ðnÞ ¼ ff0nðnÞg are introduced together with the
eigenvalue matrixΛ0= diag(λ0n). Separating stationary from non-stationary modes, the
synap-tic coupling vector ~w0 ¼ fh@ycnj�0ijy¼myg and matrixW0¼ fh@ycnj�mijy¼mygm6¼0are also
included, respectively. For the chosenσ(y), we remark that all these terms depend on the total firing rateν(t) only through μy(t).
As in [15],Eq (12)expresses the firing rateν(t) as a superposition of contributions due to both the stationary mode for a givenμy(t) (i.e., the input-output gain function F0), and the
firing rate due to non-stationary modes (~f0� ~a0) depicting how far the actual density is from the equilibrium. As a result, the farther the system from the equilibrium, the larger the contri-bution toν(t) due to the non-stationary modes. Indeed, under these conditions the expansion coefficients ~a0are significantly different from 0, whereas under stationary conditions ~a0¼ 0 andν = F0.
The first row ofEq (12)is an infinite set of equations for the infinite expansion coeffi-cients {a0n}. It describes their relaxation dynamics towards an equilibrium point which can
change with time, as the stationary one-dimensional densityϕ0depends onμy(t). Time scales
of this relaxation are dictated by the eigenvaluesΛ0and the coupling coefficients W0, the
lat-ter being 0 as the synaptic efficacyJ vanishes. Also F0, ~f0,Λ0and W0depend onμy(t) and
on the other changes in the input current received by the neurons. This means thatEq (12)
incorporates both slow and fast time scales. The former are due to the slowesta0nrelaxation
or to the synaptic low-pass filtering for largeτs, as pointed out by the second row ofEq (12).
On the other hand, fast time scales can be due to the fast changes of the input affecting directly the aforementioned equation coefficients, such as the gain function F0. This
multi-scale nature ofEq (12)makes this theoretical representation of general applicability, as shown later.
Eq (12)is one of our main results, obtained following a procedure that can be summarized as follows (seeFig 2). We described the single-neuron dynamics as a two-dimensional Lange-vinEq (1), relying on the diffusion approximation. From this, an extended mean-field approximation [25] led us to deriveEq (3), a two-dimensional continuity equation for the dynamics of the population densityρ(x, y, t). We then represented ρ as the time evolution of its projections onto the axes defined by the eigenfunctions of the FP operator, as shown in Eqs (7) and (8) – an approach similar to the Hartree-Fock method in quantum mechanics but with the main difference that here we take into account also the temporal dimension. Finally,ρ(x, y, t) is approximated assuming y ’ μy(t), which is a low-pass filtered version of
μI(t). This allows us to operate a central moment closure focusing on the instantaneous firing
rateν(t). We remark that this dimensional reduction aims at finding an effective model of
the firing rateν(t), not of the full probability density ρ(x, y, t). This means we are not expect-ing to recover accurately the full dynamics of the neuronal membrane potentialV(t), as detailed later.
Equivalence of non-instantaneous synaptic transmission and distribution of axonal delays. So far, we considered the filtering activity operated by local synapses transmitting
incoming spikes as a post-synaptic potential non-instantaneous in time. Here, we recall the known dynamics of a network of spiking neurons where synaptic transmission is instantaneous but a distribution of axonal delays is taken into account [15,27,29]. In this one-dimensional case (τs= 0), the network dynamics can be simply worked out
by replacingν in the expressions of μIand s2I (seeEq (2)) with the instantaneous rate
~
nðtÞ ¼R01nðt dÞrdðdÞdd of spikes received by neurons when a distribution ρd(δ) of axonal
transmission delaysδ is taken into account. Thus, the population density ρx(x, t) follows the
one-dimensional FP equation @trx¼ 1 tm @x ðf ðxÞ mIð~nÞ mextÞ þ 1 2 s 2 extþ s 2 Ið~nÞ � @x � � rx _~n ¼ ðn ~nÞ=td; 8 < : ð13Þ
with instantaneous firing rate n ¼ ðs2
extþ s
2
The spectral expansion of this continuity equation leads to a known firing rate equation with coefficients resulting to have a straightforward relationship with those inEq (12). Indeed, the elements of the synaptic coupling vector and matrix now are h@n~cnj�mi ¼ tmJ Ch@ycnj�mijy¼my
for anyn and m, since
~
n ¼ my
tmJ C
for what shown above. Due to this, the searched firing rate equation equivalent to the 1D FPEq (13)reduces to _ ~a ¼ ðΛ0þW0tmJ C _~nÞ~a þ ~w0tmJ C _~n _~n ¼ ðn ~nÞ=td n ¼ F0þ ~f0� ~a ; 8 > > > > < > > > > : ð14Þ
where the specific delay distribution
rdðdÞ ¼ 1 tde
d=td
YðdÞ ð15Þ
has been taken into account. This leads ~n to be a version of the collective firing rateν smoothed in time by a first-order low-pass filter with decay timeτd.Θ(δ) is the Heaviside function, as only
positive transmission delays are admitted.
A comparison between Eqs (12) and (14) remarkably points out the equivalence of having, in a spiking neuron network, a non-instantaneous synaptic transmission or a suited distribu-tion of axonal delays. InEq (14)the expansion coefficients in ~a refer to the one-dimensional densityρx(x, t), while inEq (12)~a0is only an effective representation of the two-dimensional densityρ(x, y, t). Nevertheless, provided that τs=τd, both the dynamics seen from the
perspec-tive ofν(t) are the same, as the functions F0, ~f0, ~w0, W0andΛ0are the same. In other words,
under mean-field approximation and for not too large synaptic current fluctuationsσI, having
a local synaptic filtering with cut-off frequency 1/τsis equivalent to having random axonal
delays with exponential distribution (15) and decay constantτd=τs. Although such
equiva-lence might appear as not surprising, to the best of our knowledge this is the first proof of its general validity, i.e., holding for any IF neuron model, dynamical regime and synaptic timescale.
Numerical validation of the theory
The theory (including the equivalence between Eqs(12)and(14)) developed in the previous section has been tested through extensive numerical simulations by using NEST [30] and the high-performance custom simulator implementing the event-based approach described in [31]. The parameters for the numerical simulations have been identified following the proce-dure described in the Methods section, where also the network parameters corresponding to the proposed results are summarized.
We resorted to this strategy as the dynamics represented byEq (14)is known to display a remarkable agreement between theory and simulations for spiking neuron networks with instantaneous synaptic transmission and distribution of delays [15,29,32].
Match between input-output gain functions. As a first evaluation of the effectiveness of
the low-dimensional description derived above, we inspect the simplest condition given by the asymptotic firing rate of isolated neurons with or without a synaptic filtering of the incoming
spike trains. In the absence of recurrent connectivity and under stationary condition, fromEq (12)the firing rateν is given for any τsby the single-neuron input-output gain function F0(μ,
σ). InFig 3, we show the results of this test by comparing the steady-state firing rates of LIF neurons measured from single-neuron simulations and their theoretical input-output gain function F0(μ, σ) [27,33,34]: 1 F0ðm; sÞ ¼ 2 tm Z vthr m s vres m s dy ey2 Z 1 y dz e z2 ; ð16Þ
whereμ = μI+τmIext/Cmis the total mean input current and s2 ¼ s2Iþ s
2
extis the total
vari-ance. Each curve has been obtained by changing the rateν of the external source of Poissonian spikes, such that bothμIandσIchange simultaneously, according toEq (2).
The 0-th approximation provided byEq (16)(gray dashed curves inFig 3-left) perfectly overlaps with the simulation of a single LIF neuron incorporating instantaneous synaptic transmission (τs= 0, red curves), confirming the validity of the diffusion approximation.
How-ever, the output firing rates at fixed inputμ change with increasing τs, as described in [8,11].
Such discrepancy can be better appreciated from the central panels ofFig 3, where the differ-encesΔF between F0and the gain functions Ftsfrom simulations are shown to mainly increase withτs. This could seem in contradiction with the fact that our low-dimensional
Fig 3. Match between theory and simulations in the input-output gain function of LIF neurons. Gain functions F
forN = 4000 LIF neurons (uncoupled and coupled), obtained by varying the total mean current μ = μI+τmIext/Cmfor
differentτsand different sizeσIof the synaptic current fluctuations: (A) sIjm¼vthr¼ 0:5 mV and (B) sIjm¼vthr¼ 2:0 mV.
In the explored range ofμ, σIvaries between 74% and 138% of its value atμ = vthr. Left panels in (A-B): theoretical F0 from 0-th order approximation (τs= 0, gray dashed curves) and measured Ftsforτs2 {0, 1, 4, 16} ms (reddish curves) from single-neuron simulations. Central and right panels in (A-B) show the differences DF ¼ Fts F0, either including (right) or excluding (middle) an additional unfiltered white noise with sext¼ sIjm¼vthr. Only the three curves
withτs> 0 are shown as the simulations withτs= 0 largely overlap the x-axis due to the remarkable agreement with the theoretical F0. (C) Gain functions Ftsversus the mean external current μextfor a network of excitatory neurons.
Network topology and neuron parameters as in Panel (B) withσext= 2.0 mV, but with 50% of recurrent synapses (the
others receive Poissonian spike trains as in (B)). Thick arrows: values ofμext= {15.2, 17.5, 20.5} mV driving the
network from noise- to drift-dominated regimes, respectively. Bottom panel, steady-state firing rates (fulfilling the equilibrium condition F0(ν�) =ν�) of the network with non-instantaneous synaptic transmission (blue plots) and distribution of axonal delays (red plots) at the highlightedμext. Shaded strips in all panels: mean± SEM from 10 independent simulations (seeMethodsfor further details and parameters).
reduction is independent of the synaptic filtering timescales arising in the steady-state firing rate of isolated neurons, but it is not. Indeed, by comparing two examples at relatively small and large sizeσIof the synaptic current fluctuations (Fig 3A and 3B, respectively), the
equivalence is recovered as long asσI�vthr, according to the assumption at the basis of our
0-th order approximation.
Note that the largest discrepancies are found in a relatively small range of the mean current μ around vthr, where synaptic fluctuations have a major role in the spike emission process and
this range shrinks with decreasingσI. Outside this range ofμ, our 0-th order approximation is
still valid to describe the steady-state firing rate of single-neurons both under noise- and drift-dominated regime.
Finally, we remark that the discrepancy between theory and simulations can be further reduced by adding the white noise with sizeσext> 0. We introduced this additional input in
the limitσext! 0 inEq (4)to safely manage the boundary conditions in the spectral expansion
of the one-dimensional operatorLxy. However, it has been suggested to effectively model sev-eral neuronal features like the ionic channel stochasticity [35,36] and the effect of thermal noise [37]. As a result, under this biologically plausible condition, the match between theoreti-cal input-output gain function and simulations is recovered also for relatively largeσI(Fig 3B
-right).
As a second test, we consider the same network as above, with the same total number of aver-age synaptic inputs per neuron, but with 50% of synaptic contacts from external Poisson-like excitatory neurons and 50% of recurrent synapses. Under stationary condition, we obtain results similar to those observed in the absence of recurrent connectivity:Fig 3Cshows the gain func-tions Ftsfor this network, plottedversus the mean external current μext(upper panel) and the
steady-state firing rates of the network, satisfying the equilibrium condition F(ν�) =ν�[25], plot-tedversus τs=τd(lower panel). The abscissaμextis the sum of two mean currents: the white
noise with drift and the exogenous Poissonian input. Notice that the average firing rateν0when
delay distributions are incorporated in simulation (red curves in lower panel) does not change by varyingτd. This is an expected result, as under stationary conditions the input firing rate ~n in
Eq (14)coincides with the equilibrium value independently ofτd. It is also apparent that the
mean-field equilibrium pointν�(dashed lines) mildly overestimates the simulated equilibrium pointν0when both a delay distribution (red lines) or synaptic filters (blue lines) are
incorpo-rated. This is due to the assumptions underlying the diffusion and the mean-field approxima-tions. Increasing the number of synaptic contacts per neuron and the network sizeN by keeping unchanged drift and diffusion coefficients of the related Langevin equation leads to reduce the difference |ν0− ν�|. Finally, due to the changes in the input-output gain function F0shown in
the top panel, the measured mean firing ratesν0in the network with non-instantaneous synaptic
transmission are different from those with distribution of axonal delays, and the difference increases withτs=τd[8,10,11]. This trend is particularly apparent around the critical valueμ =
vthr(see the caseμext= 17.5 mV in the bottom panel), where our 0-th order perturbative
approach is less accurate unlessσI�vthr. We remark that the equivalence between delay
distri-bution and synaptic filtering holds in both noise- (subthreshold) and drift-dominated (supra-threshold) regimes, and it is less accurate only at the transition between these two conditions.
Responses to a time-varying input. In the comparison between steady-state firing rates,
neither delay distribution nor out-of-equilibrium conditions have been explicitly tested as the input-output gain function is a single-neuron feature and the adopted input currents had sta-tionary mean and variance.
To overcome this limited testing condition, here we further test the first-order statistics of the firing rate, by simulating the network of LIF neurons shown inFig 7Awith a time-varying
input (Fig 4A). To this purpose, the rate of external spikesνext(t) was modulated as a periodic
wave with periodT = 200 ms. Average responses of the firing rate ν(t) across a stimulation period are shown inFig 4Bfor networks with the same mean-field parameters but different time scalesτd=τs. Remarkably, in all conditions, the average firing rate of each network with
synaptic filters (blue curves) widely overlaps with the one computed in the equivalent network incorporating a suited distribution of transmission delays (red curves) and also with the numerical integration (using the Python library described in [38]) of the one-dimensional FP
Eq (13)equivalent to Eqs (12) and (14). We remark that very similar results can be obtained by using only the two slowest modes of the FP operator spectrum.
The similarity of the response in the theoreticalν(t) and the two simulated networks is apparent for anyτs=τdranging from 1ms to 64 ms, during both the fast transient elicited by
Fig 4. Network response to a periodic step-wise input. (A) Example of response of a network of LIF neurons to
periodic modulation ofνext(t) = (1 + Δνexterf(2 sin(2πt/T))) 40 Hz with T = 200 ms and Δνext= 0.05 (top). Spiking
activity and firing rateν(t) (middle and bottom, respectively). Same network as inFig 7Aat drift-dominated regime with instantaneous synaptic transmission and a distribution of axonal delays withτd= 1 ms. (B) Average responses to a stimulation period (200 ms) of (blue curves) the same LIF neuron network with synaptic filters at different timescales τs= 1, 4, 16, 64 ms and (red curves) the equivalent networks with distributions of delays (τd= 1, 4, 16, 64 ms). White and gray shaded intervals mark the time when neurons in the network work in drift- (μI+μext>vthr) and
noise-dominated (μI+μext<vthr) regime, respectively. Averages are computed across 300 periods, removing a first transient
response (1 s). A grand average of thisν(t) is eventually computed across 10 networks with same mean-field parameters but randomly different connectivity matrices. Gray dashed curves: theoreticalν(t) resulting from the numerical integration of the one-dimensional FPEq (13).
the sudden changes ofνext(att = 0 and 100 ms), and the relaxation phases when the networks
drift towards a new equilibrium point. The latter effect is not a surprise, being due to the expected low-pass filtering. However, notice the reduction of the time lag in crossing 40 Hz (firing rate of the unperturbed networks) asτd=τsincreases. For networks with synaptic
filter-ing, when the input suddenly changes, the non-instantaneous synaptic transmission leads to have a uniform shift of the probability densityρx(x, t) towards the emission threshold (towards
right inFig 1). As a consequence, firing rates can display arbitrarily fast reactions, which are even faster asτsincreases [6,9]. Conversely, networks with transmission delays do not express
the same mechanism, as they have vanishing probability densities forV ! vthr. Here, the
coex-istence of slow and fast dynamics is likely due to the fact that, forτdlong enough compared
toT/2, ρx(x, t) has not enough time to approach its asymptotic value. As a consequence, just
before the input has a step transition, a fraction of neurons (those with the longest pre-synaptic transmission delays) has still memory of the previous stimulation phase. This subset of neu-rons, whose size increases withτd, are the first to be primed, thus leading the whole network to
rapidly react toνextchanges.
We remark that, asν(t) changes across the stimulation period, not only μI, but also s2I
changes accordingly. This notwithstanding, the match of the three curves is remarkably good under both drift- and noise-dominated regimes (white and gray shaded intervals inFig 4B, respectively), witnessing the quite general validity of the proposed approach.
Equivalence in the asynchronous state of finite-size networks. As the equivalence
between synaptic filter and delay distribution holds for the meanν(t), here we evaluate the equivalence validity also for the second-order statistics of the firing rate around a stable equi-librium point. To this purpose, we inspect the activity in the presence of an endogenous noise in a network of a finite numberN of neurons trapped into a stable asynchronous state. As previously done in [15], finite-size fluctuations can be incorporated into Eqs (12) and (14) as an additive forcing term to the expansion coefficient dynamics, giving rise to an equation for the firing rateνN(t) of a finite pool of neurons (seeMethodsfor details). The resulting
dynamics can be linearized around the equilibrium pointν�= F
0(ν�). This allows to compute
the Fourier transformνN(ω), and from it the power spectral density P(ω) = |νN(ω)|2, which
turns out to be PðoÞ ¼ 1 þ 2Re h ~f 0� ðioI Λ0Þ 1~ c0 i � � �1 h F00þio~f0� ðioI Λ0Þ 1 ~ w0 i rðioÞ � � � 2 n� N: ð17Þ
Here, I is the identity matrix, all the coefficients depending onν(t) are now constants computed atν(t) = ν�, F0
0 ¼ @nF0andr(iω) is the ratio between the Fourier transforms of μy
andμI, which results to ber(iω) = 1/(1 + iωτs). Note thatr(iω) coincides with the Fourier
transformρd(iω) of the delay distribution (15) withτd=τs. If we consider an additional
fixed transmission delayδ, this transform generalizes to r(iω) = exp(−iωδ)/(1 + iωτs) [15,
29].
We compared this theoretical result with theP(ω) estimated from simulations of networks composed of simple IF neurons with synaptic filtering (seeMethodsfor details), finding a remarkable agreement (Fig 5). Here we resorted to the VIF neuron model [21], an extended version of the widely used PIF neuron [39], for its amenability to analytical treatment [15] (see
Methodsfor details).
As stated above, the theoretical power spectral density has an equilibrium pointν�. On the other side, the microscopic simulations provide a firing rateνN(t) whose mean value (averaged
isν0. Each power spectral density plot provides information about the frequencies at which the
νN(t) variations are stronger.
The theoreticalP(ω) fromEq (17)is computed on the first 4096 modes/eigenfunctions of the FP operator. The large number of modes is justified by the need of capturing the network behavior over a wide frequency range, characterized by the presence of many narrow resonant peaks.
Besides the good matching between theory (solid lines) and simulations (shaded strips), it is interesting to note how resonant peaks are differently affected by increasingτs. Not
surpris-ingly, high-ω peaks are broadened due to the low-pass filtering of synaptic transmission. These resonances are related to the synaptic reverberation of spiking activity, which gives rise to the so-calledtransmission poles in the linearized dynamics [15,40]. In an excitatory network working in drift-dominated regime as inFig 5, these are expected to be found at frequencies multiple of the inverse of the average time needed by a presynaptic spike to affect postsynaptic potential, here roughly 1/(δ + τs). On the other hand, the low-ω peaks do not display any shift
in frequency, although their power is broadened as well. This is because such peaks are due to the so-calleddiffusion poles in the linearized dynamics [15,41]. They occur when neurons emit spikes at drift-dominated (suprathreshold) regime. In this regime, the distribution of the inter-spike intervals is narrow and resonant peaks emerge atω/2π multiples of ν�(equal to 10 Hz inFig 5). We remark that, due to the shifting at low-ω of the transmission peaks, a con-structive interference with diffusion peaks may occur. This explains the non-monotonic change of power of the second diffusion peak at 20 Hz inFig 5. Indeed, its power decreases whenτsis increased from 0 ms to 4 ms, as expected, whereas due to the mentioned
Fig 5. Match between theory and simulations in networks of VIF neurons working in drift-dominated regime with different synaptic decay timesτs. Power spectral densitiesP(ω) of the finite-size network firing rate νN(t) are
shown forτs= {0, 4, 32} ms. All networks have a stable equilibrium point atν�= 10 Hz, and are composed ofN = 2000
excitatory VIF neurons. Synaptic matrices are random with connection probabilityC/N = 5% and average synaptic efficacyJ = 7.510−3vthr. Spikes are delivered to the postsynaptic targets with a fixed transmission delayδ = 10 ms (see
Table 1for other parameters). Solid lines: theoreticalP(ω) fromEq (17)computed relying on the first 4096 modes of the FP operator spectral expansion. Shaded strips: mean±3 SEM (standard error mean) of P(ω) from 10 simulations with different random synaptic matrices and same mean-field parameters. Power spectra are normalized byN/ν0, whereν0is the average in time ofνN(t) and ν0=ν�for theoreticalP(ω).
interference it is heightened forτs= 32 ms. This is because the lowest transmission peak in this
case is expected to be found atω/2π = 23.8 Hz, not too far from 2ν�= 20 Hz.
Equivalence under noise- and drift-dominated regime. To further test the equivalence,
we investigated whether this match worked well for networks of neurons not only under drift-dominated spiking regime, as shown inFig 5, but also under noise-driven regimes. Indeed, synaptic filtering may have significantly different effects on the steady-state firing rate in these two regimes [9,11], as shown inFig 3.
Fig 6Ashows for comparison the power spectral densitiesP(ω) of the firing rate ν(t) from an excitatory VIF neuron network working under noise-dominated regime by varyingτs(=
τd), in which either non-instantaneous synaptic transmission (blue curves) or a distribution of
transmission delays (red curves) was incorporated. The theoreticalP(ω) fromEq (17)(gray dashed curves) overlaps the simulation results for a wide range ofω. Notice that we used the first 256 modes of the FP operator spectrum, i.e., 16 times less modes than for the drift-domi-nated case: this points out the greater effectiveness of our approach in the noise-domidrift-domi-nated regime. It is also important to remark that the discrepancy observable at low-ω is not due to a Fig 6. Equivalence of non-instantaneous synaptic transmission and distribution of synaptic delays in a network of VIF neurons. (A) Power spectral densitiesP(ω) of ν(t) obtained for τs(=τd) = {0, 4, 32} ms in an excitatory VIF neuron networks under noise-dominated regime with either non-instantaneous synaptic transmission (blue) or distribution of transmission delays (red). Gray dashed curves: theoreticalP(ω) fromEq (17)computed relying on the first 256 modes of the FP operator spectral expansion. (B) Power spectral densityPs(ω) of the simulated network activity in the case of
non-instantaneous synaptic transmission, varying the synaptic time constantτs. Dashed lines:P(ω) sections plotted in Fig 5and in panel A. (C)Pd(ω) for networks with an exponential distribution of spike transmission delays and with
instantaneous synaptic transmission, varying the delay time constantτd. In B and C, top and bottom panels show the results for the network set in an asynchronous state in which neurons are in a drift- (top) and noise-dominated (bottom) regime, respectively. (D) Iso-power curves from panels B (solid lines) and C (dashed lines). (E) Average firing rateν0from the simulations used for the left panels with non-instantaneous synaptic transmission (blue plots) and distribution of axonal delays (red plots). Error bars representing SEM are not visible. The not specified network parameters are as inFig 5(seeTable 1). Also here, power spectra are averages across 10 simulations with different random synaptic matrices and same mean-field parameters.
failure of our approximation, as it occurs also atτs=τd= 0 ms, when synaptic transmission is
instantaneous. This indeed is related to the mean-field approximation, which overestimates the firing rate of the equilibrium point, as explained more in detail later in this subsection.
Fig 6B(non-instantaneous synaptic transmission) andFig 6C(distribution of transmission delays) show the same comparisons for a larger set ofτs=τdvalues (in logarithmic scale),
cor-responding to networks working in noise- (bottom panels) or drift-dominated (top panels) regimes; in this case, the power spectrum amplitude is color-coded. The power spectral densi-ties in panel A correspond to the cuts marked by dashed horizontal lines in (bottom) panels B and C. A remarkable agreement between simulations in the two tested conditions is confirmed byFig 6D, showing iso-power curves from panels B (solid lines) and C (dashed lines): the equivalence of non-instantaneous synaptic transmission and distribution of axonal delays is witnessed by the tight overlapping of solid and dashed iso-power curves for a wide range of fil-ter timescales.
The same good matching between these two network types is apparent also under noise-dominated regime (bottom panels). As expected in this regime, the diffusion poles become real numbers [15] and resonant peaks at frequencies multiple (byω/2π) of the equilibrium fir-ing rateν�disappear, although the high-ω resonances corresponding to transmission poles
remain almost unaffected by this change of regime. We point out that widening the filtering time window (i.e., increasingτsandτd) leads to almost flat power spectra (seeFig 5),
compati-bly with the flattening of the response amplitude of isolated neurons receiving colored noise as input currents [9,10].
As shown inFig 1B–1D, the section ofρ(x, y, t) close to firing threshold vthrshrinks whenτs
increases from 4 ms to 64 ms. In our perturbative approach this dependence is completely neglected, as the distribution is assumed to be a Dirac’sδ.
We remark that the shown comparisons rely on normalized spectra, i.e.P(ω)N/ν0. On the
one hand, this means that power spectra shapes do not depend on the particular transmission protocol implemented in the network. On the other hand, this does not guarantee, as shown for the LIF neuron case, that average firing ratesν0are the same and do not change with the
protocol-related timescales. To address this issue, inFig 6Ethe measuredν0are compared,
showing features completely similar to those observed inFig 3for the LIF neuron model. Notice that the discrepancy between theory and simulations (ν0andν�, respectively)
explains the differences found at low-ω inFig 6A. Indeed, such difference leads to have differ-ent slopes of the gain function (F0
0ðn0Þ 6¼ F 0 0ðn
�Þ), which in turn affect differently the power
spectrum at low-ω as Pðo ¼ 0Þ / ð1 F0
0Þ 2
.
Since we already checked the accuracy of the theoreticalP(ω), in the rest of the paper we will focus on the power spectral densities estimated from microscopic simulations. Taking as reference these simulations instead ofEq (14)in the comparison with the supposedly equiva-lent networks with synaptic filtering has indeed the advantage of allowing to test our theoreti-cal prediction also in neuron models for which analytitheoreti-cal expressions for the eigenvalues and eigenfunctions of the FP operator are not available, as in the case of exponential IF (EIF, [42]) neurons. Moreover, this strategy allowed us to overcome the possible issue related to comput-ing a large number of modes of the FP operator spectrum for more realistic scomput-ingle-cell models.
Independence from spiking neuron models. The developed theory is of general
applica-bility, i.e. it applies to a wide range of spiking neuron models. As a result, the equivalence proved for a network of simplified VIF neurons together with the theoretical expression for the power spectra ofν(t) are both expected to hold also for networks of more realistic single-cell models like the LIF and the EIF neurons.
To verify this expectation, we directly simulated networks of LIF and EIF neurons with a mean-field stable equilibrium point atν�= 40 Hz. InFig 7the meanP(ω)N/ν
0for these kinds
of networks is compared both under drift-dominated (top panels) and noise-dominated (bot-tom) regimes. A remarkable agreement is apparent for both LIF and EIF neuron networks (Fig 7A and 7B, respectively), confirming the generality of the developed approach. To further test the absence of any bias due to the specific neuron model chosen, we computed the relative differences between the power spectral densities when non-instantaneous synaptic filtering (Ps) and transmission delay distribution (Pd) are taken into account. The cumulative
distribu-tions of these discrepancies across all tested time scales (τs=τd) and Fourier frequencies (ω/
2π) are shown inFig 7C. Interestingly, no significant differences are visible for the three cho-sen models (VIF, LIF, EIF), further proving that in all regimes (drift- and noise-dominated) our dimensional reduction does not depend on the specific single neuron dynamics.
Starting from this, it is not surprising to note here that spectral features similar to those highlighted inFig 6for VIF neuron networks are also displayed by LIF and EIF neuron net-works. More specifically, resonant peaks at multiple frequencies ofν0under drift-dominated
regimes and resonances at higher-ω due to the transmission poles—i.e., those related to the average spike transmission delay—are also visible in bothFig 7A and 7Bin LIF and EIF neu-ron networks, respectively.
Equivalence beyond the asynchronous state. So far, we tested the validity of the
equiva-lence between non-instantaneous synaptic transmission and delay inhomogeneity in lineariz-able dynamical regimes like the asynchronous state and in purely excitatory networks. To further assess the generality of our theoretical results, we simulated a multi-modular network Fig 7. Equivalence between synaptic filtering and delay distribution in networks with different types of IF neurons. Match between relative power spectral densitiesP(ω)N/ν0in networks of leaky (A) and exponential (B) integrate-and-fire excitatory neurons (LIF and EIF, respectively) with non-instantaneous synaptic transmission (solid lines) and distributions of synaptic delays (dashed lines). Network parameters are set in order to have an asynchronous state with average firing rateν0= 40 Hz. Contour lines as inFig 6D.P(ω) are averages across 10 simulations with different random synaptic matrices and same mean-field parameters. (C) Cumulative distribution of relative differencesΔP(ω) = Pd(ω)/Ps(ω) − 1 across different time scales (τsandτd) and Fourier frequenciesω from panels A and B for LIF (red) and EIF (orange) neuron networks, respectively. As a reference, also the same cumulative distribution for VIF neuron networks (obtained fromP(ω) inFig 6D) is plotted (black).ΔP(ω) are measured as z-scores, i.e. averages divided by standard deviations of values across simulations. Top and bottom panels: networks working under drift-dominated and noise-dominated regime, respectively (see parameters in Tables2and3).
containing an inhibitory (I) and an excitatory (E) population of LIF neurons (Fig 8A, see
Methods), with the purpose of analyzing its dynamical behaviors by changing a bifurcation parameter.
In this specific example network, the bifurcation parameter is the percentagecEI=cII�cαI
of outward inhibitory synaptic connections per neuron. To compensate for these changes in the inhibitory component, the synaptic efficacies of both recurrent and external excitatory neurons are progressively increased: this allows to keep an equilibrium pointν�= F
0(ν�) at the
same relatively high firing rate n�
E¼ n
�
I ¼ 10 Hz for both populations, regardless of its stability
(see details inMethods). For low enoughcαI, this equilibrium point is unstable and only an
asynchronous irregular state at low firing rate (denoted as AL) can be reached by both popula-tions, as shown inFig 8B-bottom for the E-I network withcαI= 10% and non-instantaneous
synaptic transmission. A suited increase ofcαIstabilizes the high-frequency asynchronous state
(denoted as AH), in which the neurons fire at the equilibrium rate of 10 Hz (Fig 8B-middle, cαI= 50%). A stronger synaptic inhibition eventually leads the E-I network to display
synchro-nous firing with coherent global oscillations (GO) of the spiking activity (Fig 8B-top,cαI=
90%).
We compared networks with both synaptic filtering and delay distribution across this rep-ertoire of dynamical regimes, by making extensive simulations on a finer grid ofcαIvalues, by
using the same mean-field parameters (Fig 8C). In all cases the networks undergo the same state transitions at the same criticalcαIvalues. More specifically, the valuecαI� 12.5% marks a
Fig 8. Comparison between multi-modular networks of inhibitory and excitatory LIF neurons with either synaptic filtering or transmission delay distribution. (A) Sketch of the multi-modular network. (B) Three examples of firing rate time evolution for each network stable state. Plum and orange
curves represent inhibitory and excitatory neuronal populations, respectively, with synaptic filter. (C) Left plots: mean (να0,α = E, I) and standard
deviation (σν) of the instantaneous firing rate across time (binning size of 0.1 ms, discarding the first second of the time series) for inhibitory (bottom
panels) and excitatory (top panels) populations. Right plots: match between relative power spectral densitiesP(ω)N/να0for excitatory (top panel) and
inhibitory (bottom panel) populations with non-instantaneous synaptic transmission (solid lines) and distributions of transmission delays (dashed lines). Contour lines as inFig 6D. Different states are obtained by increasing the connection probability with pre-synaptic inhibitory neurons, namely cαI�cEI=cII(see text for details). The spectraP(ω) are averages across 10 simulations with different random synaptic matrix realizations. Dashed
horizontal lines roughly mark the bifurcations. Rightmost arrows point out the values ofcαIcorresponding to the examples displayed in panels B and D.
(D) Comparisons between relative spectra from networks with delay distribution (red lines) and synaptic filtering (blue lines) for the three network states AL (dotted lines), AH (dashed-dotted lines), and GO (solid lines), for excitatory (top panel) and inhibitory (bottom panel) populations.
transcritical bifurcation where the low-frequency and the 10-Hz equilibrium points exchange their stability and the transition from AL to AH occurs. In both networks, the mean firing rate ν0of the AL state increases with its standard deviationσν(Fig 8C-left) until the 10-Hz
equilib-rium point corresponding to the AH state becomes the lowest stable equilibequilib-rium point. This point undergoes a supercritical Hopf bifurcation atcαI� 70%, where the network has a
transi-tion from the AH to the GO state. The resulting synchronous oscillatransi-tions of the firing rate have an amplitude that increases withcαI. This is whyσνincreases andν0remains unchanged
at 10 Hz. Even in this case, the first two moments of the firing rate in networks with both non-instantaneous synaptic transmission (blue) and distribution of axonal delays (red) match almost perfectly.
This close resemblance is also apparent in the relative power spectraP(ω)N/ν0shown inFig
8C-right, although significant changes in the power distribution emerge when the network state has a transition. To better appreciate the comparison between non-instantaneous synap-tic transmission (solid lines) and distributions of axonal delays (dashed lines),Fig 8D(where cαI= {10, 50, 90}%, as inFig 8B) shows three paradigmatic relative spectral densities. As
expected,P(ω) for the network in the AL state close to the bifurcation (red and blue dotted curves) displays a large peak at low-ω due to the slow fluctuations of the firing rate determined by the weakening of the 10-Hz attractor stability. AsPðo ¼ 0Þ ’ ð1 F0
0Þ 1
(seeEq (17)), the small but significant differences between the spectra at low-ω for non-instantaneous synaptic transmission (blue) and distributions of axonal delays (red) have to be attributed to the differ-ent shape of the input-output gain functions F0. Indeed, our approximated F0overestimates
the steady-state firing rate whenτs> 0 (seeFig 3) in a relatively narrow range of the mean
input currentμ around vthr, thus affecting also the slope F 0
0. This difference is also responsible for the small discrepancies between the measuredν0andν�as inFig 6E. Further increasing the
recurrent inhibitioncαI, in the AH state (dashed-dotted lines) resonance peaks due to
trans-mission poles atω/2π � 50 Hz pop up. In the GO state (solid lines) also higher-order harmon-ics start to be visible in bothPd(ω) and Ps(ω). Although the overlap between these spectra
is remarkable also in the GO regime, a small unexpected shift of the resonant frequencies is visible, highlighting a mild increase of the oscillation frequency when the synaptic filter is incorporated.
Similar results were obtained for the same network topology by changing the filtering time-scales, as shown inFig 9.Fig 9Ashows relative power spectral densities for the inhibitory pool with non-instantaneous synaptic transmission. As expected, increasingτsfrom 1 ms to 16 ms
(from bottom to top panels, respectively) makes the network more stable, thus shifting at highercαIthe critical value at which the transition to the GO state occurs and the peaks
associ-ated to higher-order harmonics pop up. Largerτsalso imply a reduction of the pace of the
damped and synchronous oscillations fromω/2π � 90 Hz to � 40 Hz.
The cumulative distributions of theZ-score ΔP(ω) = Pd(ω)/Ps(ω) − 1 inFig 9Bshow, for
the three considered timescales (τs=τd), that the differences of spectra display always the
same statistics for both excitatory (orange) and inhibitory (blue) populations, similarly to
Fig 7C. This equivalence under these conditions is also apparent by comparing the coeffi-cient of variationcv= std(ν)/mean(ν) = σν/ν0for the excitatory and inhibitory pools (top
and bottom panels inFig 9C, respectively), with non-instantaneous synaptic transmission (blue) and exponential delay distributions (red). These plots confirm and complete the results already described inFig 8Cforτs=τd= 4 ms: for lowcαI(network in the AL state)
the relative large variability is mainly due to a reduction ofν0; moreover,cvremains at its
lowest value until the transition from AH to GO occurs. After this bifurcation, it increases with the amplitude of the global oscillation. As remarked above, the Hopf bifurcation
